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Finite-Length Analysis of Polar Secrecy Codes for Wiretap Channels

Hessam Mahdavifar, Fariba Abbasi

TL;DR

The paper addresses finite-length secrecy performance of polar secrecy codes for degraded wiretap channels, deriving a new MIS leakage lower bound and combining it with the standard upper bound to obtain non-asymptotic MIS guarantees. By leveraging Hassani's finite-length scaling results for polar codes, it bounds the secrecy-gap C_s - R_s and the normalized MIS leakage, showing decay rates of O(n^{-1/μ̄*}) and O(n^{-1/μ̄}), respectively, and shows how to operate slightly above the secrecy capacity by removing a small randomness subset to achieve a gap of n^{-1/δ}. The results provide practical, non-asymptotic secrecy guarantees and a numerical framework to bound MIS leakage at finite blocklengths, enabling design guidance for polar secrecy codes in real systems. Overall, the work advances non-asymptotic physical-layer secrecy by connecting polar code scaling, MIS leakage, and finite-length performance for wiretap channels, with actionable constructions and numerical validation.

Abstract

In a classical wiretap channel setting, Alice communicates with Bob through a main communication channel, while her transmission also reaches an eavesdropper Eve through a wiretap channel. In this paper, we consider a general class of polar secrecy codes for wiretap channels and study their finite-length performance. In particular, bounds on the normalized mutual information security (MIS) leakage, a fundamental measure of secrecy in information-theoretic security frameworks, are presented for polar secrecy codes. The bounds are utilized to characterize the finite-length scaling behavior of polar secrecy codes, where scaling here refers to the non-asymptotic behavior of both the gap to the secrecy capacity as well as the MIS leakage. Furthermore, the bounds are shown to facilitate characterizing numerical bounds on the secrecy guarantees of polar secrecy codes in finite block lengths of practical relevance, where directly calculating the MIS leakage is in general infeasible.

Finite-Length Analysis of Polar Secrecy Codes for Wiretap Channels

TL;DR

The paper addresses finite-length secrecy performance of polar secrecy codes for degraded wiretap channels, deriving a new MIS leakage lower bound and combining it with the standard upper bound to obtain non-asymptotic MIS guarantees. By leveraging Hassani's finite-length scaling results for polar codes, it bounds the secrecy-gap C_s - R_s and the normalized MIS leakage, showing decay rates of O(n^{-1/μ̄*}) and O(n^{-1/μ̄}), respectively, and shows how to operate slightly above the secrecy capacity by removing a small randomness subset to achieve a gap of n^{-1/δ}. The results provide practical, non-asymptotic secrecy guarantees and a numerical framework to bound MIS leakage at finite blocklengths, enabling design guidance for polar secrecy codes in real systems. Overall, the work advances non-asymptotic physical-layer secrecy by connecting polar code scaling, MIS leakage, and finite-length performance for wiretap channels, with actionable constructions and numerical validation.

Abstract

In a classical wiretap channel setting, Alice communicates with Bob through a main communication channel, while her transmission also reaches an eavesdropper Eve through a wiretap channel. In this paper, we consider a general class of polar secrecy codes for wiretap channels and study their finite-length performance. In particular, bounds on the normalized mutual information security (MIS) leakage, a fundamental measure of secrecy in information-theoretic security frameworks, are presented for polar secrecy codes. The bounds are utilized to characterize the finite-length scaling behavior of polar secrecy codes, where scaling here refers to the non-asymptotic behavior of both the gap to the secrecy capacity as well as the MIS leakage. Furthermore, the bounds are shown to facilitate characterizing numerical bounds on the secrecy guarantees of polar secrecy codes in finite block lengths of practical relevance, where directly calculating the MIS leakage is in general infeasible.
Paper Structure (10 sections, 22 equations, 2 figures)

This paper contains 10 sections, 22 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation Convention
  4. Polar Codes
  5. Polar Secrecy Codes
  6. Bounds on the MIS Leakage of Polar Secrecy Codes
  7. Finite-Length Scaling of Polar Secrecy Codes
  8. Bounds on the Scaling Exponent
  9. Operating above the Secrecy Capacity
  10. Numerical Results

Figures (2)

  • Figure 1: Wyner's wiretap channel setting.
  • Figure 2: Upper and lower Bounds versus the Monte-Carlo Approximation of the normalized MIS leakage over wiretap BEC$(\epsilon)$.

Theorems & Definitions (3)

  • proof
  • proof
  • proof